TITLE: 

           SHAPE OPTIMIZATION BY THE HOMOGENIZATION METHOD

AUTHORS:
      
           G.ALLAIRE (Commissariat a l'Energie Atomique
                      DRN/DMT/SERMA, C.E. Saclay
                      91191 Gif sur Yvette, France
                     & Laboratoire d'Analyse Num\'erique, 
                       Universit\'e Paris 6)

           E.BONNETIER (CMAP)

           G.FRANCFORT (Institut Galilee
                        Universite Paris-Nord
                        93430 Villetaneuse, France)

           F.JOUVE (CMAP)

ABSTRACT:

In the context of shape optimization, we seek minimizers of the sum of the  
elastic compliance and of the weight of a solid structure under specified 
loading. This problem is known not to be well-posed, and a relaxed 
formulation is introduced. Its effect is to allow for microperforated 
composites as admissible designs. In a two-dimensional setting the relaxed 
formulation was obtained in [6] with the help of the theory of 
homogenization and optimal bounds for composite materials. We 
generalize the result to the three dimensional case. Our contribution 
is twofold; first, we prove a relaxation theorem, valid in any dimensions;
secondly, we introduce a new numerical algorithm for computing optimal designs, 
complemented with a penalization technique which permits to remove composite 
designs in the final shape. Since it places no assumption on the number of 
holes cut within the domain, it can be seen as a topology optimization 
algorithm. Numerical results are presented for various two and three 
dimensional problems.


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